A small sphere (emissivity = 0.661, radius= r 1 ) is located at the center of as

Question

A small sphere (emissivity = 0.661, radius= r1) is located at the center of aspherical asbestos shell (thickness = 1.16 cm, outer radius= r2; thermal conductivity of asbestos is0.090 J/(s m Co)). The thickness of the shell is smallcompared to the inner and outer radii of the shell. The temperatureof the small sphere is 587 °C, while the temperature of theinner surface of the shell is 421 °C, both temperaturesremaining constant. Assumingthat r2/r1 = 10.4and ignoring any air inside the shell, find the temperature indegrees Celsius of the outer surface of the shell.

Explanation / Answer

   the heat Q conducted during a time t through a bar oflength L and cross sectional area a is given
   by
   Q = k A2 T t / L
   where k is the thermal conductivity of asbestos
   A2 is the area of the spherical shell
   A2 = r22
   T = (421.0oC - T2)
   L is thickness of the shell
   on substituting we get T2 as
   T2 = (Q L / k (r22) t) + 421.0oC
   the net radiant energy is given by
   Q = P t
       = e A1(T4 - To4) t
   where e is emmisivity
   is stefan boltzman constant
   A1 is spherical area of the sphere
   A1 = r12
   T is the temperature of the sphere given by
   T = 587.0oC
      = 860 K
   To = 873.2 K
   so on substitution T2 gives
   T2 = 421oC - ([e(T4 - To4)] L / k(r2 / r1)2)
        = ..............oC

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